Expanding from endorin briggs answer with some thoroughly amateur arm-chair physics and assumptions. For reference, the theory is that a Red Giant could account for Quo'Nos having a green sky.
According to Wikipedia's article on Diffuse Sky Radiation, it seems our sky shows light blue because short wavelengths (blue-ish) scatter more than longer wavelengths (reddish). This is apparently why the sky appears more yellow around the sun, as the light is traveling more directly towards you, and thus is subject to less scattering.
Our sun has a pretty broad color spectrum, although it cuts off in the violets, otherwise our sky would probably look much more purple.
The emissions from our sun peak in the "visible spectrum", which includes our standard colors. A red giant's emissions peak in the infrared, but would also pass into the visible part of the spectrum, just at much lower intensity, and probably tapering out with very low concentrations into blue/violet. If Quo'nos orbited a Red Giant, the light would have less radiation in the blue or violet wavelengths than it did in the green wavelengths.
Now, it would still have much more red light, but depending on how the radiation curve looked, we could conceivably end up with enough radiation in the green range to scatter more (especially if Quo'nos has a thicker atmosphere) green light, but have very little blue light left to overpower it. Red scatters the least of these, so the green could theoretically be the dominant color. If this were true we would also expect to see a strong reddish halo or region in the sky around the sun itself. Interestingly, the horizon might look blueish at noon on Quo'Nos, as that's the location where the most blue/violet light scattering would be visible.
Now, if we step back to the TV show, we can see that the Klingon's favor deep red lighting in general, and tend to have darker (to human eyes) illumination aboard their ships/stations. This could indicate that they see further into the infrared spectrum than humans. If so, this would support them evolving as predators on a planet orbiting a Red-Giant, where their eyes would be adapted to the different color spectrum.